Minimal equation
Minimal equation
Simplified equation
| $y^2 = x^5 + 2x^3 + 2x$ | (homogenize, simplify) |
| $y^2 = x^5z + 2x^3z^3 + 2xz^5$ | (dehomogenize, simplify) |
| $y^2 = x^5 + 2x^3 + 2x$ | (homogenize, minimize) |
Invariants
| Conductor: | \( N \) | \(=\) | \(262144\) | \(=\) | \( 2^{18} \) |
|
| Discriminant: | \( \Delta \) | \(=\) | \(524288\) | \(=\) | \( 2^{19} \) |
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(26\) | \(=\) | \( 2 \cdot 13 \) |
| \( I_4 \) | \(=\) | \(-2\) | \(=\) | \( -2 \) |
| \( I_6 \) | \(=\) | \(40\) | \(=\) | \( 2^{3} \cdot 5 \) |
| \( I_{10} \) | \(=\) | \(2\) | \(=\) | \( 2 \) |
| \( J_2 \) | \(=\) | \(208\) | \(=\) | \( 2^{4} \cdot 13 \) |
| \( J_4 \) | \(=\) | \(1888\) | \(=\) | \( 2^{5} \cdot 59 \) |
| \( J_6 \) | \(=\) | \(-2304\) | \(=\) | \( - 2^{8} \cdot 3^{2} \) |
| \( J_8 \) | \(=\) | \(-1010944\) | \(=\) | \( - 2^{8} \cdot 11 \cdot 359 \) |
| \( J_{10} \) | \(=\) | \(524288\) | \(=\) | \( 2^{19} \) |
| \( g_1 \) | \(=\) | \(742586\) | ||
| \( g_2 \) | \(=\) | \(129623/4\) | ||
| \( g_3 \) | \(=\) | \(-1521/8\) |
Automorphism group
| \(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2$ |
|
| \(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $D_4$ |
|
Rational points
Number of rational Weierstrass points: \(2\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z \oplus \Z/{2}\Z\)
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + xz + 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2\) | \(0.970076\) | \(\infty\) |
| \((0 : 0 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + xz + 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2\) | \(0.970076\) | \(\infty\) |
| \((0 : 0 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + xz + 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(1/2xz^2\) | \(0.970076\) | \(\infty\) |
| \((0 : 0 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
BSD invariants
| Hasse-Weil conjecture: | unverified |
| Analytic rank: | \(1\) |
| Mordell-Weil rank: | \(1\) |
| 2-Selmer rank: | \(2\) |
| Regulator: | \( 0.970076 \) |
| Real period: | \( 7.048010 \) |
| Tamagawa product: | \( 2 \) |
| Torsion order: | \( 2 \) |
| Leading coefficient: | \( 3.418554 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | Root number* | L-factor | Cluster picture | Tame reduction? |
|---|---|---|---|---|---|---|---|
| \(2\) | \(18\) | \(19\) | \(2\) | \(-1^*\) | \(1\) | no |
Galois representations
For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.
For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.
| Prime \(\ell\) | mod-\(\ell\) image | Is torsion prime? |
|---|---|---|
| \(2\) | 2.90.3 | yes |
| \(3\) | 3.270.1 | no |
Sato-Tate group
| \(\mathrm{ST}\) | \(\simeq\) | $J(E_4)$ |
| \(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over the number field \(\Q (b) \simeq \) 4.2.2048.1 with defining polynomial:
\(x^{4} - 2\)
Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
\(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
\(g_4 = 160 b^{2} - 96\)
\(g_6 = -1792 b^{3} + 2304 b\)
Conductor norm: 16384
\(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
\(g_4 = 160 b^{2} - 96\)
\(g_6 = 1792 b^{3} - 2304 b\)
Conductor norm: 16384
Endomorphisms of the Jacobian
Not of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
| \(\End (J_{})\) | \(\simeq\) | \(\Z\) |
| \(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
| \(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R\) |
Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) 8.0.16777216.2 with defining polynomial \(x^{8} - 4 x^{6} + 8 x^{4} - 4 x^{2} + 1\)
Not of \(\GL_2\)-type over \(\overline{\Q}\)
Endomorphism ring over \(\overline{\Q}\):
| \(\End (J_{\overline{\Q}})\) | \(\simeq\) | a non-Eichler order of index \(4\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\) |
| \(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\mathrm{M}_2(\)\(\Q\)\()\) |
| \(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\mathrm{M}_2 (\R)\) |
Remainder of the endomorphism lattice by field
Over subfield \(F \simeq \) \(\Q(\sqrt{-1}) \) with generator \(\frac{2}{3} a^{6} - \frac{7}{3} a^{4} + \frac{14}{3} a^{2} - \frac{4}{3}\) with minimal polynomial \(x^{2} + 1\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{-1}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-1}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) \(\Q(\sqrt{2}) \) with generator \(-\frac{1}{3} a^{6} + \frac{2}{3} a^{4} - \frac{1}{3} a^{2} - \frac{4}{3}\) with minimal polynomial \(x^{2} - 2\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) \(\Q(\sqrt{-2}) \) with generator \(a^{6} - 4 a^{4} + 7 a^{2} - 2\) with minimal polynomial \(x^{2} + 2\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 4.2.2048.1 with generator \(\frac{4}{3} a^{7} - \frac{14}{3} a^{5} + \frac{25}{3} a^{3} - \frac{5}{3} a\) with minimal polynomial \(x^{4} - 2\):
| \(\End (J_{F})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, not simple
Over subfield \(F \simeq \) 4.0.2048.1 with generator \(-a^{7} + 4 a^{5} - 8 a^{3} + 3 a\) with minimal polynomial \(x^{4} + 2\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{2}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{2}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 4.0.2048.1 with generator \(\frac{2}{3} a^{7} - \frac{7}{3} a^{5} + \frac{11}{3} a^{3} + \frac{2}{3} a\) with minimal polynomial \(x^{4} + 2\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{2}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{2}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 4.2.2048.1 with generator \(\frac{1}{3} a^{7} - \frac{5}{3} a^{5} + \frac{10}{3} a^{3} - \frac{8}{3} a\) with minimal polynomial \(x^{4} - 2\):
| \(\End (J_{F})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, not simple
Over subfield \(F \simeq \) \(\Q(\zeta_{8})\) with generator \(\frac{1}{3} a^{6} - \frac{5}{3} a^{4} + \frac{10}{3} a^{2} - \frac{5}{3}\) with minimal polynomial \(x^{4} + 1\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{-1}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-1}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple